In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring \(M_k(R)\) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring \(M_k(R)\) are one sided ideals in the group matrix ring \(M_k(R)G\) and the corresponding codes over the ring R are \(G^k\)-codes of length kn. Additionally, we give a generator matrix for self-dual codes, which consist of the mentioned above matrix construction. We employ this generator matrix to search for binary self-dual codes with parameters [72, 36, 12] and find new singly-even and doubly-even codes of this type. In particular, we construct 16 new Type I and 4 new Type II binary [72, 36, 12] self-dual codes.

在这项工作中，我们研究由组矩阵环中的元素生成的代码。我们提出了一种矩阵构造，用于在两个不同的环境空间中生成代码：矩阵环\（M_k（R）\）和环R，其中R是可交换的Frobenius环。我们证明环\（M_k（R）\）上的代码是群矩阵环\（M_k（R）G \）中的单边理想，并且环R上的相应代码为\（G ^ k \） -长度kn的代码。另外，我们给出了用于自对偶代码的生成器矩阵，该矩阵由上述矩阵构造组成。我们使用此生成器矩阵来搜索参数为[72、36、12]的二进制自对偶代码，并找到这种类型的新的单双和双双代码。特别是，我们构造了16个新的I型和4个新的II型二进制[72、36、12]自对偶代码。